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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2604.15254 |
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| _version_ | 1866910135998218240 |
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| author | García-Sánchez, Enrique |
| author_facet | García-Sánchez, Enrique |
| contents | These notes are a detailed, self-contained introductory course on convexity and concavity in Banach lattices, suitable for both experts and beginners. We revisit, from a modern perspective, the classical notions of $(p,q)$-convexity, $(p,q)$-concavity and upper and lower $p$-estimates, and the main relations between these properties, integrating more recent developments in the area. We explain in full detail the $p$-convexification and $p$-concavification techniques and how they can be used to build renormings of Banach lattices that improve the convexity and concavity constants. We also provide a comprehensive exposition of the main factorization results for $(p,q)$-convex and $(p,q)$-concave operators, including well-known results from Krivine, Maurey--Nikishin, Pietsch and Pisier, and their applications to the representation of convex and concave Banach lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15254 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convexity and concavity in Banach lattices García-Sánchez, Enrique Functional Analysis 46B42 (primary), 46B20, 46E30, 47B60, 47A68 (secondary) These notes are a detailed, self-contained introductory course on convexity and concavity in Banach lattices, suitable for both experts and beginners. We revisit, from a modern perspective, the classical notions of $(p,q)$-convexity, $(p,q)$-concavity and upper and lower $p$-estimates, and the main relations between these properties, integrating more recent developments in the area. We explain in full detail the $p$-convexification and $p$-concavification techniques and how they can be used to build renormings of Banach lattices that improve the convexity and concavity constants. We also provide a comprehensive exposition of the main factorization results for $(p,q)$-convex and $(p,q)$-concave operators, including well-known results from Krivine, Maurey--Nikishin, Pietsch and Pisier, and their applications to the representation of convex and concave Banach lattices. |
| title | Convexity and concavity in Banach lattices |
| topic | Functional Analysis 46B42 (primary), 46B20, 46E30, 47B60, 47A68 (secondary) |
| url | https://arxiv.org/abs/2604.15254 |